NMOF (version 2.10-0)

mvFrontier: Computing Mean--Variance Efficient Portfolios

Description

Compute mean--variance efficient portfolios and efficient frontiers.

Usage

mvFrontier(m, var, wmin = 0, wmax = 1, n = 50, rf = NA,
           groups = NULL, groups.wmin = NULL, groups.wmax = NULL)
mvPortfolio(m, var, min.return, wmin = 0, wmax = 1, lambda = NULL,
            groups = NULL, groups.wmin = NULL, groups.wmax = NULL)

Value

For mvPortfolio, a numeric vector of weights.

For mvFrontier, a list of three components:

return

returns of portfolios

volatility

volatilities of portfolios

weights

A matrix of portfolio weights. Each column holds the weights for one portfolio on the frontier. If rf is specified, an additional row is added, providing the cash weight.

The i-th portfolio on the frontier corresponds to the i-th elements of return and

volatility, and the i-th column of

portfolio.

Arguments

m

vector of expected returns

var

expected variance--covariance matrix

wmin

numeric: minimum weights

wmax

numeric: maximum weights

n

number of points on the efficient frontier

min.return

minimal required return

rf

risk-free rate

lambda

risk--reward trade-off

groups

a list of group definitions

groups.wmin

a numeric vector

groups.wmax

a numeric vector

Author

Enrico Schumann

Details

mvPortfolio computes a single mean--variance efficient portfolio, using package quadprog. It does so by minimising portfolio variance, subject to constraints on minimum return and budget (weights need to sum to one), and min/max constraints on the weights.

If \(\lambda\) is specified, the function ignores the min.return constraint and instead solves the model $$\min_w\ \ -\lambda \mbox{\code{m}}'w + (1-\lambda) w'\mbox{\code{var}\,}w$$ in which \(w\) are the weights. If \(\lambda\) is a vector of length 2, then the model becomes $$\min_w\ \ -\lambda_1 \mbox{\code{m}\,}'w + \lambda_2 w'\mbox{\code{var}\,}w$$ which may be more convenient (e.g. for setting \(\lambda_1\) to 1).

mvFrontier computes returns, volatilities and compositions for portfolios along an efficient frontier. If rf is not NA, cash is included as an asset.

References

Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. tools:::Rd_expr_doi("10.1016/C2017-0-01621-X")

Schumann, E. (2023) Financial Optimisation with R (NMOF Manual). https://enricoschumann.net/NMOF.htm#NMOFmanual

See Also

minvar for computing the minimum-variance portfolio

Examples

na <- 4
vols <- c(0.10, 0.15, 0.20,0.22)
m <- c(0.06, 0.12, 0.09, 0.07)
const_cor <- function(rho, na) {
    C <- array(rho, dim = c(na, na))
    diag(C) <- 1
    C
}
var <- diag(vols) %*% const_cor(0.5, na) %*% diag(vols)

wmax <- 1          # maximum holding size
wmin <- 0.0          # minimum holding size
rf <- 0.02

if (requireNamespace("quadprog")) {
  p1 <- mvFrontier(m, var, wmin = wmin, wmax = wmax, n = 50)
  p2 <- mvFrontier(m, var, wmin = wmin, wmax = wmax, n = 50, rf = rf)
  plot(p1$volatility, p1$return, pch = 19, cex = 0.5, type = "o",
       xlab = "Expected volatility",
       ylab = "Expected return")
  lines(p2$volatility, p2$return, col = grey(0.5))
  abline(v = 0, h = rf)
} else
  message("Package 'quadprog' is required")